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\title{A Note on Tall Cardinals and Level by Level Equivalence
\thanks{2010 Mathematics Subject Classifications:
03E35, 03E55.}
\thanks{Keywords: Supercompact cardinal, strongly compact cardinal,
strong cardinal,
tall cardinal, indestructibility,
Magidor iteration of Prikry forcing, Gitik iteration of forcings
satisfying the Prikry property.
%Easton support iteration of Prikry type forcings.
}}
%Gitik iteration}}
\author{Arthur W.~Apter\thanks{The
author's research was partially
supported by PSC-CUNY grants.}\\
Department of Mathematics\\
Baruch College of CUNY\\
New York, New York 10010 USA\\
and\\
The CUNY Graduate Center, Mathematics\\
365 Fifth Avenue\\
New York, New York 10016 USA\\
http://faculty.baruch.cuny.edu/aapter\\
awapter@alum.mit.edu}
%\date{April 19, 2015}
%\date{June 16, 2015}
%\date{June 30, 2015}
\date{July 4, 2015}
%\date{\today}
\begin{document}
\maketitle
%\newpage
%\vfill\eject
\begin{abstract}
Starting from a model $V \models ``$ZFC + GCH + $\gk$
is supercompact + No cardinal is supercompact up to a measurable cardinal'',
we force and construct a model $V^\FP$ such that
$V^\FP \models ``$ZFC + $\gk$ is
supercompact + No cardinal is supercompact up to a measurable cardinal +
$\gd$ is measurable iff $\gd$ is tall'' in which level by level equivalence between
strong compactness and supercompactness holds. This extends and generalizes
both \cite[Theorem 1]{AG14} and the results of \cite{AS97a}.
\end{abstract}
\baselineskip=24pt
\section{Introduction and Preliminaries}\label{s1}
We begin with some definitions and terminology.
Suppose $\gk$ is a cardinal and $\gl \ge \gk$ is
an arbitrary ordinal. $\gk$ is {\em $\gl$ tall} if
there is an elementary embedding $j : V \to M$
with critical point $\gk$ such that $j(\gk) > \gl$
and $M^\gk \subseteq M$. $\gk$ is {\em tall} if $\gk$
is $\gl$ tall for every ordinal $\gl$.
Hamkins made a systematic study of tall cardinals in \cite{H09}.
Suppose $V$ is a model of ZFC
containing at least one supercompact cardinal
in which for all regular cardinals
$\gk \le \gl$, $\gk$ is $\gl$ strongly
compact iff $\gk$ is $\gl$ supercompact.
%except possibly if $\gk$ is a measurable limit
%of cardinals $\gd$ which are $\gl$ supercompact.
Such a model will be said to
witness {\em level by level
equivalence between strong
compactness and supercompactness}.
%The exception is provided by a theorem of Menas \cite{Me},
%who showed that if $\gk$ is a measurable limit of cardinals
%$\gd$ which are $\gl$ supercompact, the $\gk$ is $\gl$
%strongly compact but need not be $\gl$ supercompact.
We will also say that {\em $\gk$
is a witness to level by
level equivalence between strong
compactness and supercompactness}
iff for every regular cardinal $\gl \ge \gk$,
$\gk$ is $\gl$ strongly compact
iff $\gk$ is $\gl$ supercompact.
Models in which level by level
equivalence between strong compactness
and supercompactness holds nontrivially
were first constructed in \cite{AS97a}.
For brevity, we will henceforth write only
{\em level by level equivalence}.
Turning now to the main narrative,
in \cite{AG14}, the following theorem was proven.
\begin{theorem}\label{t1}{\bf (\cite[Theorem 1]{AG14})}
Suppose $V \models ``$ZFC + GCH + $\gk$ is
supercompact + No cardinal $\gl > \gk$ is measurable''.
There is then a partial ordering $\FP \in V$ such that
$V^\FP \models ``$ZFC + $\gk$ is supercompact + No cardinal $\gl > \gk$
is measurable + $\gd$ is measurable iff $\gd$ is tall''.
\end{theorem}
\noindent Theorem \ref{t1} extends and generalizes \cite[Corollary 4.3]{H09}.
The purpose of this note is to %generalize Theorem \ref{t1} and
show that it is possible to combine the results of Theorem \ref{t1} with
the property of level by level equivalence.
%{\em level by level equivalence between strong compactness
%and supercompactness} introduced by the author and Shelah in \cite{AS97a}.
Specifically, we will prove the following theorem.
\begin{theorem}\label{t2}
Suppose $V \models ``$ZFC + GCH + $\gk$ is
supercompact + No cardinal is supercompact up to a measurable cardinal''.
There is then a partial ordering $\FP \in V$ such that
$V^\FP \models ``$ZFC + $\gk$ is supercompact + No cardinal is supercompact
up to a measurable cardinal + $\gd$ is measurable iff $\gd$ is tall''.
In $V^\FP$, level by level equivalence holds.
\end{theorem}
We remark that the hypotheses of Theorem \ref{t2} automatically
imply the hypotheses of Theorem \ref{t1}. This is since otherwise,
if there were a measurable cardinal above the supercompact
cardinal $\gk$, $\gk$ and by reflection, unboundedly in $\gk$
many cardinals below it, would be supercompact up to a measurable cardinal.
Also, Theorem \ref{t2} may be thought of as a
very strong ``identity crisis'' type of result, the study of which
was first initiated by Magidor in \cite{Ma76}.
In the model witnessing the conclusions of Theorem \ref{t2},
not only do the measurable and tall cardinals coincide precisely,
but in addition, the partially strongly compact and partially supercompact
cardinals also coincide precisely, as long as the coincidence is at regular levels.
Further, the unique supercompact cardinal $\gk$ in the model witnessing the
conclusions of Theorem \ref{t2} is also the unique strongly compact cardinal.
This is since any strongly compact cardinal $\gd < \gk$ would of necessity
have to witness a failure of level by level equivalence,
%between strong compactness and supercompactness,
and there are no measurable cardinals above $\gk$.
%We begin with the following definition.
%\begin{definition}\label{d1}
%{\bf (Hamkins \cite{H09})}
%Suppose $\gk$ is a cardinal and $\gl \ge \gk$ is
%an arbitrary ordinal. $\gk$ is {\em $\gl$ tall} if
%there is an elementary embedding $j : V \to M$
%with critical point $\gk$ such that $j(\gk) > \gl$
%and $M^\gk \subseteq M$. $\gk$ is {\em tall} if $\gk$
%is $\gl$ tall for every ordinal $\gl$.
%\end{definition}
Before beginning the proofs of our theorems, we briefly mention
some preliminary information and terminology.
%Before beginning the proofs of
%Theorems \ref{t1} and
%\ref{p1}, we briefly
%mention some preliminary information.
Essentially, our notation and terminology are standard, and
when this is not the
case, this will be clearly noted.
For $\ga < \gb$ ordinals, $[\ga, \gb]$,
$[\ga, \gb)$, $(\ga, \gb]$, and $(\ga, \gb)$
are as in the usual interval notation.
%If $\gk \ge \go$ is a regular cardinal and $\gl$
%is an arbitrary ordinal, then
%$\add(\gk, \gl)$ is the standard partial ordering
%for adding $\gl$ Cohen subsets of $\gk$.
When forcing, $q \ge p$ will mean that
{\it $q$ is stronger than $p$}.
If $G$ is $V$-generic over $\FP$,
we will abuse notation slightly and
use both $V[G]$ and $V^{\FP}$
to indicate the universe obtained by forcing with $\FP$.
If $x \in V[G]$, then $\dot x$ will be a term in $V$ for $x$.
We may, from time to time, confuse terms with the sets they denote
and write $x$
when we actually mean $\dot x$
or $\check x$, especially
when $x$ is some variant of the generic set $G$, or $x$ is
in the ground model $V$.
The abuse of notation mentioned above will be compounded
by writing $x \in V^\FP$ instead of $\dot x \in V^\FP$.
%Any term for trivial forcing will
%always be taken as a term for the
%partial ordering $\{\emptyset\}$.
If $\varphi$ is a formula in the forcing language
with respect to $\FP$ and $p \in \FP$, then
$p \decides \varphi$ means that
{\it $p$ decides $\varphi$}.
As in \cite{GS}, we will say that
the partial ordering $\FP$
is {\em $\gk^+$-weakly closed
and satisfies the Prikry property} if
it meets the following criteria.
\begin{enumerate}
\item $\FP$ has two partial
orderings $\le$ and $\le^*$ with
$\le^* \ \subseteq \ \le$.
\item For every $p \in \FP$
and every statement $\varphi$
in the forcing language
with respect to $\FP$, there
is some $q \in \FP$ such that
$p \le^* q$ and $q \decides \varphi$.
%($q$ decides $\varphi$).
\item The partial ordering
$\le^*$ is $\gk$-closed, i.e.,
there is an upper bound for every
increasing chain of conditions having length $\gk$.
\end{enumerate}
\noindent Note that if $\FP$ is $\gk^+$-weakly closed and
satisfies the Prikry property, then in analogy to Prikry forcing,
$V$ and $V^\FP$ contain the same subsets of $\gk$.
From time to time within the course of our
discussion, we will refer to
partial orderings $\FP$ as being
{\it Gitik iterations of forcings satisfying the Prikry property}.
%{\it Easton support iterations of Prikry type forcings}.
By this we will mean an Easton support iteration
as first given by Gitik in \cite{G86},
to which we refer readers for a discussion
of the basic properties of
and terminology associated with such an iteration.
Key to the proofs of Theorems \ref{t1} and \ref{t2} is
the following result due to Gitik and
Shelah \cite{GS}. It is a corollary of
the work of \cite[Section 2]{GS} and is an analogue to
\cite[Theorem 9]{AG14}.
\begin{theorem}\label{t3}
Suppose $V \models ``$ZFC + GCH +
$\gk$ is a strong cardinal greater than $\gd$''.
%$\gd < \gk$ are such that $\gd$ is a regular cardinal and $\gk$ is a strong cardinal''.
Let $\gd^*$ be the least measurable cardinal greater than $\gd$.
There is then a $(\gd^*)^+$-weakly closed partial ordering $\FI(\gd, \gk)$
satisfying the Prikry property having cardinality $\gk$
%which first acts nontrivially on an ordinal greater than %or equal to $\gd^*$
such that $V^{\FI(\gd, \gk)} \models ``\gk$ is a strong cardinal whose
strongness is indestructible under $\gk^+$-weakly
closed partial orderings satisfying the Prikry property''.
\end{theorem}
We mention that we are assuming some
familiarity with the large cardinal
notions of measurability,
%measurable cardinals of high Mitchell order, hypermeasurability,
tallness, strongness, strong compactness, and supercompactness.
Interested readers may consult \cite{J}
or \cite{H09}. %\cite{SRK}. %for further details.
We note only that we will say {\em $\gk$ is supercompact
(strongly compact) up to the cardinal $\gl$} if $\gk$ is
$\gd$ supercompact (strongly compact) for every $\gd < \gk$.
\section{The Proof of Theorem \ref{t2}}\label{s2}
We turn now to the proof of Theorem \ref{t2}.
\begin{proof}
Suppose $V \models ``$ZFC + GCH + $\gk$ is supercompact +
No cardinal is supercompact up to a measurable cardinal''.
Without loss of generality, by first doing the forcing of \cite{AS97a},
we abuse notation and also assume that level by level equivalence
%between strong compactness an supercompactness
holds in $V$.
Let ${\mathcal C} = \{\gd < \gk \mid \gd$ is a strong
cardinal which is not a limit of strong cardinals$\}$.
$\FI$ will be the partial ordering of \cite[Lemma 2.2]{AG14}
defined using Theorem \ref{t3} which makes each $\gd \in {\mathcal C}$
%$V$-strong cardinal $\gd$ which is not also a limit of $V$-strong cardinals
a strong cardinal
indestructible under $\gd^+$-weakly closed partial orderings satisfying the Prikry property.
We explicitly give the definition of $\FI$ now, quoting almost verbatim from \cite{AG14}.
Let $\la \gd_\ga \mid \ga < \gk \ra$ enumerate in increasing order the
members of ${\cal C}$. %together with their limit points.
For every $\ga < \gk$, let $\gg_\ga = \sup_{\gb < \ga} \gd_\gb$,
where $\gg_0 = \go$.
%where $\gg_\ga = \go$ if $\ga = 0$.
%Let $\la \gg_\ga \mid \ga < \gk \ra$ enumerate in
%increasing order $\{\go\}$ together with the successors of
%the limit points of members of ${\cal C}$.
$\FI = \la \la \FI_\ga, \dot \FQ_\ga \ra \mid \ga < \gk \ra$
is taken as the
%Gitik style Easton support iteration of
Gitik iteration of forcings satisfying the Prikry property %Prikry type forcings of
of length $\gk$ such that $\FI_0 = \{\emptyset\}$.
For every $\ga < \gk$,
$\FI_{\ga + 1} = \FI_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for the partial ordering
%which, at any stage $\ga < \gk$, forces
$\FI(\gg_\ga, \gd_\ga)$ of Theorem \ref{t3} as defined in
$V^{\FI_\ga}$. Note that this makes sense,
since inductively, it is the case that
$\card{\FI_\ga} < \gd_\ga$.
By the Hamkins-Woodin results \cite{HW},
$V^{\FI_\ga} \models ``\gd_{\ga}$ is a strong cardinal'',
meaning that $\FI_{\ga + 1}$ as just given is valid.
%Note also that the only strong cardinals on which
%$\FI$ acts nontrivially are those strong cardinals
%which are not limits of strong cardinals in $V$.
%In other words, if $V \models ``\gd$ is a strong
%cardinal which is a limit of strong cardinals'', then
%$\FI$ acts trivially on $\gd$.
\begin{lemma}\label{l1}
%Suppose $V \models ``$Either $\gd \in {\mathcal C}$
%or $\gd$ is a measurable limit of members of ${\mathcal C}$''.
Suppose that %in $V$,
%either $\gd \in {\mathcal C}$, or
$V^\FI \models ``\gd$ is a measurable limit of members of ${\mathcal C}$''.
Then $V^\FI \models ``$Level by level equivalence
holds at $\gd$''.
\end{lemma}
\begin{proof}
%Suppose first that $\gd \in {\mathcal C}$, with $\gd = \gd_\ga$ for some $\ga < \gk$.
Let $\gd$ be as in the hypotheses for Lemma \ref{l1}.
%, with $\ga$ such that either $\gd = \gd_\ga$ if $\gd \in {\mathcal C}$, or $\gd = \gg_\ga$ if $\gd$ is a %measurable limit of members of ${\mathcal C}$.
Write $\FI = \FI_\gd \ast \dot \FI^\gd$.
%where $\FI_\gd$
%$\FI_\gd$ is $\FI$ defined
%through stage $\gd$, and $\dot \FI^\gd$ is a term for the remainder of $\dot \FI$.
%acts nontrivially on ordinals below $\gd$, and $\dot \FI^\gd$ is a term for
%the rest of $\FI$. Thus, by the definition of $\FI$, $\dot \FI^\gd$ is
%forced to act nontrivially on ordinals above $\gd$.
%We emphasize that in general, because of the definition
%of $\FI$ given above, {\rm it is not the case that
%$\FI^*_\gd = \FI_\gd$}.
%By \cite[Lemma 2.2]{AG14}, if $\gd \in {\mathcal C}$, then $\gd$ remains strong and hence
%If $\gd \in {\mathcal C}$, then $\gd$ is strong and therefore Mahlo in $V$.
Because $V^\FI \models ``\gd$ is a measurable limit of members of ${\mathcal C}$'',
$\gd$ is %measurable in $V^\FI$ and consequently
Mahlo in $V^\FI$. Since
forcing can't create a new Mahlo cardinal, $\gd$ is Mahlo in $V$.
Thus, by the definition of $\FI$,
$\FI_\gd$ is the direct limit of $\la \FI_\ga \mid \ga < \gd \ra$.
%regardless if $\gd \in {\mathcal C}$ or
%$V^\FI \models ``\gd$ is a measurable limit of members of ${\mathcal C}$'',
%a direct limit is taken at stage $\gd$ in the definition of $\FI$.
Hence, $\FI_\gd$ is $\gd$-c.c., $\card{\FI_\gd} = \gd$, $\gg_\gd = \gd$, and
$\forces_{\FI_\gd} ``$ Forcing with $\dot \FI^\gd$ adds no bounded subsets of $(\gd^*)^V =
(\gd^*)^{V^{\FI_\gd}}$''.
%(So if $\gd \in {\mathcal C}$, $\FI_\gd$ will be $\gd$-c.c$.$ and will act on ordinals below
%$\gd$ and force $\gd$ to be a strong cardinal whose strongness is indestructible
%under $\gd^+$-weakly closed partial orderings satisfying the Prikry property.
%If $\gd$ is a measurable limit of members of ${\mathcal C}$,
Suppose now $\gl \ge \gd$ is such that $V^\FI \models ``\gd$ is $\gl$
strongly compact and $\gl$ is regular''. We first assume that $\gl \in [\gd, (\gd^*)^V)$.
Since $\forces_{\FI_\gd} ``\dot \FI^\gd$ adds no bounded subsets of $(\gd^*)^V$'',
it follows that $\forces_{\FI_\gd} ``\gd$ is $\gl$ strongly compact''.
Note that since $\gd$
is measurable in $V^{\FI_\gd}$,
$\gd$ is once again Mahlo in $V^{\FI_\gd}$.
%and thus also Mahlo in $V$. Consequently, as we observed above,
%$\FI_\gd$ is the direct limit of
%$\la \FI_\ga \mid \ga < \gd \ra$, and
%$\FI_\gd$ satisfies $\gd$-c.c$.$ in $V$.
This means that
since $\FI_\gd$ satisfies
$\gd$-c.c$.$ in $V^{\FI_\gd}$ as well
(this follows because $\gd$ is Mahlo in
$V^{\FI_\gd}$ and $\FI_\gd$
is a subordering of the
direct limit of
$\la \FI_\ga \mid \ga < \gd \ra$
as calculated in $V^{\FI_\gd}$),
(the proof of)
\cite[Lemma 8]{A97} (see in particular
the argument found starting in
\cite[third paragraph of page 111]{A97}) or (the proof of)
\cite[Lemma 3]{AC1} tells us that every $\gd$-additive
uniform ultrafilter over a regular cardinal
$\gb \ge \gd$ present in
$V^{\FI_\gd}$ must be an extension
of a $\gd$-additive uniform ultrafilter
over $\gb$ in $V$.
Therefore, since the $\gl$
strong compactness of $\gd$ in
$V^{\FI_\gd}$ implies
%by Ketonen's theorem of \cite{Ke}
that every
$V^{\FI_\gd}$-regular cardinal
$\gb \in [\gd, \gl]$ carries
a $\gd$-additive uniform ultrafilter
in $V^{\FI_\gd}$,
and since the fact $\FI_\gd$ is $\gd$-c.c$.$
%is the direct limit of $\la \FI_\ga \mid \ga < \gd \ra$
tells us the regular cardinals
at or above $\gd$ in
$V^{\FI_\gd}$ are the same
as those in $V$,
the preceding sentence implies
that every $V$-regular cardinal
$\gb \in [\gd, \gl]$ carries a
$\gd$-additive uniform ultrafilter
in $V$.
Ketonen's theorem of \cite{Ke}
then implies that
$\gd$ is $\gl$ strongly
compact in $V$.
%By Ketonen's criterion [???] for strong compactness, this means that
%in $V^{\FI_\gd}$, every regular $\gg \in [\gd, \gl]$
%carries a $\gd$-additive, uniform ultrafilter. %$\U_\gd$.
%Because $\FI_\gd$ is $\gd$-c.c.,
%the regular cardinals at and above $\gd$ in $V^{\FI_\gd}$ are the same as
%the regular cardinals at and above $\gd$ in $V$.
%the regular cardinals in the interval $[\gd, \gl]$ in $V^{\FI_\gd}$
%are the same as %those in $V$.
%the regular cardinals in the interval $[\gd, \gl]$ in $V$.
%Further, the argument of [???] (see also [???]) shows that because
%$\FI_\gd$ is $\gd$-c.c., every regular $\gg \in [\gd, \gl]$ carries in $V$ a
%$\gd$-additive, uniform ultrafilter. Ketonen's criterion from [???]
%once again shows that $V \models ``\gd$ is $\gl$ strongly compact''.
By level by level equivalence, $V \models ``\gd$ is $\gl$ supercompact'' as well.
Let $G$ be $V$-generic over ${\FI}_\gd$.
Take $j : V \to M$ as an elementary embedding witnessing the
$\gl$ supercompactness of $\gd$ generated by a supercompact
ultrafilter over $P_\gd(\gl)$. Write
$j(\FI_\gd) = \FI_\gd \ast \dot \FI'$. Because ${\rm cp}(j) = \gd$,
$j({\cal C}) \cap \gd = {\cal C} \cap \gd$.
Further, if %$V \models ``\gg < \gd$ is a strong cardinal'',
$V \models ``\gg$ is a member of ${\cal C} \cap \gd$'',
$M \models %``j(\gg) = \gg$ is a strong cardinal''.
``j(\gg) = \gg$ is a member of $j({\cal C}) \cap \gd = {\cal C} \cap \gd$''.
Thus, since
$V \models ``\gd$ is a Mahlo limit of members of ${\mathcal C} \cap \gd$''
and $M^\gl \subseteq M$,
%(all of which are strong cardinals)'',
$M \models ``\gd$ is a Mahlo limit of %strong cardinals''.
members of $j({\cal C}) \cap \gd = {\cal C} \cap \gd$''.
By the definition of $\FI$ and $\FI_\gd$, this means that in $M$,
$\forces_{\FI_\gd} ``\dot \FI'$ is $((\gd^*)^+)^M$-weakly closed and
satisfies the Prikry property''. Also, as $\gl < (\gd^*)^V$, $\gl < (\gd^*)^M$.
This is since otherwise, if $(\gd^*)^M \le \gl$, then because $M^\gl \subseteq M$,
$(\gd^*)^M$ is measurable in both $V$ and $M$. Again using the fact
$M^\gl \subseteq M$, it follows that in $V$ as well as $M$, $\gd$ is supercompact
up to the measurable cardinal $(\gd^*)^M$. This is a contradiction to our
hypotheses on $V$. In particular, in both $V$ and $M$, as $\gl < (\gd^*)^M$,
$\forces_{\FI_\gd} ``\dot \FI'$ is $\gl^+$-weakly closed and satisfies the Prikry property''.
As in the proof of \cite[Lemma 2.2]{AG14},
we may now apply the argument of \cite[Lemma 1.5]{G86}.
We again feel free to quote almost verbatim from \cite{AG14} as appropriate.
Specifically, since GCH in $V$ implies that
$V \models ``2^{\gl} = \gl^+$'', we may let
$\la \dot x_\ga \mid \ga < \gl^+ \ra$ be an
enumeration in $V$ of all of the
canonical ${\FI_\gd}$-names of subsets of
$(P_\gd(\gl))^{V[G]}$.
Because $\FI_\gd$ is $\gd$-c.c$.$ and $M^\gl \subseteq M$,
$M[G]^\gl \subseteq M[G]$.
By \cite[Lemmas 1.4 and 1.2]{G86}, we may therefore
define in $V[G]$ an increasing sequence
$\la p_\ga \mid \ga < \gl^+ \ra$
of elements of $j({\FI_\gd})/G$
such that if $\ga < \gb < \gl^+$,
$p_\gb$ is an
Easton extension of $p_\ga$,\footnote{Roughly
speaking, this means that
$p_\gb$ extends $p_\ga$ as in a usual
reverse Easton iteration, except that
at coordinates at which, e.g., %something like
Prikry forcing or some variant or
generalization thereof occurs in $p_\ga$,
measure 1 sets are shrunk and stems are not
extended. For a more precise definition,
readers are urged to consult \cite{G86}.
We do note, however, that $\le^*$ here is Easton extension.}
every initial segment of
the sequence is in $M[G]$, and for every
$\ga < \gl^+$,
$p_{\ga + 1} \decides
``\la j(\gb) \mid
\gb < \gl \ra \in j(\dot x_\ga)$''.
The remainder of the argument of
\cite[Lemma 1.5]{G86} remains valid and
shows that a supercompact ultrafilter
${\cal U}$ over ${(P_\gd(\gl))}^{V[G]}$
may be defined in $V[G]$ by
$x \in {\cal U}$ iff
$x \subseteq {(P_\gd(\gl))}^{V[G]}$ and
for some $\ga < \gl^+$ and some
${\FI_\gd}$-name $\dot x$ of
$x$, in $M[G]$, $p_\ga \forces_{j({\FI_\gd})/G}
``\la j(\gb) \mid
\gb < \gl \ra \in j(\dot x)$''.
(The fact that $j '' G = G$ tells
us ${\cal U}$ is well-defined.)
%is a supercompact ultrafilter in $V[G]$ over $P_\gk(\gl)$.
Thus, $\forces_{{\FI_\gd}} ``\gd$ is $\gl$ supercompact'',
so since $\forces_{\FI_\gd} ``$Forcing with $\dot \FI^\gd$
adds no bounded subsets of $(\gd^*)^V$'',
$V^{\FI_\gd \ast \dot \FI^\gd} = V^\FI \models ``\gd$ is $\gl$ supercompact''.
%This means that
The work of the preceding paragraph shows that
if $\gl \ge \gd$ is such that $\gl \in [\gd, (\gd^*)^V)$ and
$V^\FI \models ``\gd$ is $\gl$ strongly compact and $\gl$ is regular'', then
$V^\FI \models ``\gd$ is $\gl$ supercompact''.
The proof of Lemma \ref{l1} will therefore be completed by showing that
for no $V^\FI$-regular cardinal $\gl \ge (\gd^*)^V$ is it true that
$V^\FI \models ``\gd$ is $\gl$ strongly compact''.
Assume to the contrary that %$V^\FI \models ``\gd$ is $\gl$ strongly compact''.
$\gl$ is a counterexample. Since $V^\FI \models ``\gd$ is $\gl$ strongly compact'',
$\gl \ge (\gd^*)^V$, and $\forces_{\FI_\gd} ``$Forcing with $\dot \FI^\gd$
adds no bounded subsets of $(\gd^*)^V$'', $V^{\FI_\gd} \models ``\gd$ is
strongly compact up to $(\gd^*)^V$''.
Because the regular cardinals at and above $\gd$
are the same in both $V$ and $V^{\FI_\gd}$, this means that
for any ($V$ or $V^{\FI_\gd}$)-regular cardinal $\gl \in (\gd, (\gd^*)^V)$,
$V^{\FI_\gd} \models ``\gd$ is $\gl$ strongly compact''.
By our arguments in the second paragraph of the proof of this lemma,
$V \models ``\gd$ is $\gl$ supercompact''.
This, however, is a contradiction to our assumption that
$V \models ``$No cardinal is supercompact up to a measurable cardinal'', since
$(\gd^*)^V$ is the least measurable cardinal greater than $\gd$ in $V$.
This contradiction completes the proof of Lemma \ref{l1}.
\end{proof}
%The above shows that if in $V$, either $\gd \in {\mathcal C}$,
%or $\gd$ is a measurable limit of members of ${\mathcal C}$,
%and $V^\FI \models ``\gl \ge \gd$ is regular and $\gd$ is $\gl$
%strongly compact'', then $V^\FI \models ``\gd$ is $\gl$ supercompact''.
%This yields that if $\gd$ remains measurable in $V^\FI$, then in fact,
%level by level equivalence holds at $\gd$. To complete the proof of Lemma \ref{l1},
%it therefore suffices to show that $\gd$ remains measurable in $V^\FI$.
%We note that the last paragraph of the proof of Lemma \ref{l1} shows that if
\begin{lemma}\label{l2}
If %either $\gd \in {\mathcal C}$, or
$V^\FI \models ``\gd$ is a measurable limit of members of ${\mathcal C}$'', then
$V^\FI \models ``\gd$ is not supercompact up to a measurable cardinal''.
\end{lemma}
\begin{proof}
%We again use the notation of the proof of Lemma \ref{l1}.
Suppose to the contrary $\gg \ge \gd$ is such that
$V^\FI \models ``\gd$ is supercompact up to $\gg$ and $\gg$ is measurable''.
We have that
$\forces_{\FI_\gd} ``$Forcing with $\dot \FI^\gd$ adds no bounded subsets of
$(\gd^*)^V = (\gd^*)^{V^{\FI_\gd}}$''.
Consequently, $V^\FI \models ``\gg \ge (\gd^*)^V$'', and
$V^{\FI_\gd} \models ``\gd$ is supercompact (and hence also
strongly compact) up to $(\gd^*)^V$''.
The last paragraph of the proof of Lemma \ref{l1} shows that this is a contradiction.
This completes the proof of Lemma \ref{l2}.
\end{proof}
We continue as in the proof of \cite[Theorem 1]{AG14}.
We assume now that our ground model is $V^\FI$.
%, which with an abuse of notation we relabel as $V$,
%has the properties of the model $V^\FI$ constructed above.
Given this, and adopting the notation of \cite[Theorem 1]{AG14},
let $\FP(\gg_\ga, \gd_\ga)$ for every $\ga < \gk$
be the Magidor iteration of Prikry forcing from \cite{Ma76}
%as defined in $V$
which adds a Prikry
sequence to every measurable cardinal in
the open interval $(\gg_\ga, \gd_\ga)$.
The partial ordering
$\FP = \la \la \FP_\ga, \dot \FQ_\ga \ra \mid \ga < \gk \ra$
with which we force
is defined as the
%Gitik style Easton support iteration of
Gitik iteration of forcings satisfying the Prikry property of
length $\gk$ such that $\FP_0 = \{\emptyset\}$.
For every $\ga < \gk$,
$\FP_{\ga + 1} = \FP_\ga \ast \dot \FQ_\ga$, where
$\dot \FQ_\ga$ is a term for the partial ordering
%which, at any stage $\ga < \gk$, forces
$\FP(\gg_\ga, \gd_\ga)$ %of Lemma \ref{l1}
as defined in $V^\FI$ (and {\em not as defined in $V^{\FI \ast \dot \FP_\ga}$}, which means that
$\FP$ may be equivalently defined in $V^\FI$ as the Easton support product
$\prod_{\ga < \gk} \FP(\gg_\ga, \gd_\ga)$ --- see \cite[line 3 of the proof of Lemma 2.3]{AG14}).
By \cite[Lemmas 2.1 -- 2.4]{AG14} and the intervening remarks,
$V^{\FI \ast \dot \FP} \models ``\gk$ is supercompact + $\gd < \gk$
is measurable iff $\gd$ is tall iff either $\gd \in {\mathcal C}$ or
$\gd$ is a measurable limit of members of ${\mathcal C}$''.
Because $V \models ``$No cardinal is supercompact up to a measurable
cardinal + $\gk$ is supercompact'', $V \models ``$No cardinal above
$\gk$ is measurable''. %Because $\FI \ast \dot \FP$ may be defined so that
Since by its definition, $\card{\FI \ast \dot \FP} = \gk$, by the
L\'evy-Solovay results \cite{LS}, $V^{\FI \ast \dot \FP} \models ``$No
cardinal above $\gk$ is measurable''. Hence, $V^{\FI \ast \dot \FP} \models ``\gd$
is measurable iff $\gd$ is tall iff either $\gd \in {\mathcal C}$ or $\gd$
is a measurable limit of members of ${\mathcal C}$''.
The proof of Theorem \ref{t2} will therefore be complete once we have shown the following.
\begin{lemma}\label{l3}
In $V^{\FI \ast \dot \FP}$, no cardinal is supercompact up to a measurable cardinal, and
level by level equivalence holds.
\end{lemma}
\begin{proof}
We first note that these facts are true if
$V^{\FI \ast \dot \FP} \models ``\gd$ is a measurable limit of members of ${\mathcal C}$''.
To see this, working in $V^\FI$, write $\FP = \FP_\gd \ast \dot \FP^\gd$.
%where as before, $\FP_\gd$ acts nontrivially on ordinals below $\gd$,
%$\FP$ defined through stage $\gd$, and $\dot \FP^\gd$ is a term for the rest of $\FP$.
Because $\gd$ is measurable and hence Mahlo in
$V^{\FI \ast \dot \FP}$ and forcing can't create a new Mahlo cardinal, $\gd$ is Mahlo in $V^\FI$.
Therefore, by the definition of $\FP$,
$\FP_\gd$ is the direct limit of $\la \FP_\ga \mid \ga < \gd \ra$, so as in
the proof of Lemma \ref{l1}, $\FP_\gd$ satisfies $\gd$-c.c$.$ in both
$V^\FI$ and $V^{\FI \ast \dot \FP_\gd}$. Further, by
\cite[page 351, paragraph immediately prior to Lemma 2.3]{AG14},
$\forces_{\FP_\gd} ``$Forcing with $\dot \FP^\gd$ adds no bounded subsets of
$(\gd^*)^{V^{\FI \ast \dot \FP_\gd}} = (\gd^*)^{V^\FI}$''. This means that the arguments
of Lemma \ref{l1} remain valid with $\FI$ in $V$ replaced by $\FP$ in $V^\FI$ and show that
if $V^{\FI \ast \dot \FP} \models ``\gd$ is a measurable limit of members
of ${\mathcal C}$'', then level by level equivalence
holds at $\gd$ in $V^{\FI \ast \dot \FP}$. The arguments of
Lemma \ref{l2} also remain valid and show that if
$V^{\FI \ast \dot \FP} \models ``\gd$ is a measurable limit of
members of ${\mathcal C}$'', then $\gd$ is not supercompact
up to a measurable cardinal in $V^{\FI \ast \dot \FP}$.
We consider now what happens if $\gd \in {\mathcal C}$.
Under these circumstances, we will show that $V^{\FI \ast \dot \FP} \models
``\gd$ is not $\gd^+$ strongly compact''.
This automatically implies that level by level equivalence holds at $\gd$,
%since as we have already observed,
since $\gd$ is measurable iff $\gd$ is
$\gd$ strongly compact iff $\gd$ is $\gd$ supercompact.
%To see this, as in the preceding paragraph,
%working in $V^\FI$, write $\FP = \FP_\gd \ast \dot \FP^\gd$.
%where $\FP_\gd$ acts nontrivially on ordinals below $\gd$,
%and $\dot \FP^\gd$ is a term for the rest of $\FP$.
%Again as in the preceding paragraph,
%As we have already observed,
%$\forces_{\FP_\gd} ``$Forcing with $\dot \FP^\gd$ adds no bounded subsets of
%$(\gd^*)^{V^{\FI \ast \dot \FP_\gd}} = (\gd^*)^{V^\FI}$''.
%It therefore suffices to show that
%$V^{\FI \ast \dot \FP_\gd} \models ``\gd$ is not $\gd^+$ strongly compact''.
To see this, assume towards a contradiction that
$V^{\FI \ast \dot \FP} \models ``\gd$ is $\gd^+$ strongly compact''.
By the fact
$\FP$ may be equivalently defined as the Easton support product
$\prod_{\ga < \gk} \FP(\gg_\ga, \gd_\ga)$, it is possible in $V^\FI$ to write
$\FP = \FQ^0 \times \FQ^1 \times \FQ^2$.
Here, $\card{\FQ^0} < \gd$, $\FQ^1 = \FP(\gg, \gd)$,
where $\gg = \gg_\gb$ for the ordinal $\gb$ such that $\gd = \gd_\gb$, and
$\FQ^2 = \prod_{\ga \ge \gb + 1} \FP(\gg_\ga, \gd_\ga)$.
Since $\card{\FQ^0} < \gd$, the results of \cite{LS} tell us that
$V^{\FI \ast (\dot \FQ^1 \times \dot \FQ^2)} \models ``\gd$ is $\gd^+$ strongly compact''.
Again by \cite[page 351, paragraph immediately prior to Lemma 2.3]{AG14},
$\forces_{\FI} ``$Forcing with $\dot \FQ^2$ adds no bounded subsets of
$(\gd^*_\gb)^{V^\FI}$''. Thus, $\FQ^1$ has the same definition in both
$V^\FI$ and $V^{\FI \ast \dot \FQ^2}$, and $V^{\FI \ast \dot \FQ^1} \models ``\gd$
is $\gd^+$ strongly compact''.
%Ketonen's criterion of \cite{Ke} then implies that
Hence, both $\gd$ and
$\gd^+$ carry $\gd$-additive, uniform ultrafilters in $V^{\FI \ast \dot \FQ^1}$.
However, because the Magidor iteration of Prikry forcing from \cite{Ma76}
does not collapse any cardinals, by \cite[Theorem 3.1]{Ma76} and its proof,
both $\gd$ and $\gd^+$ carry $\gd$-additive, uniform ultrafilters in $V^\FI$.
Ketonen's theorem of \cite{Ke} therefore again implies that
$V^\FI \models ``\gd$ is $\gd^+$ strongly compact''.
Write $\FI = \FI^1 \ast \dot \FI^2$, where $\FI^1$ is the
portion of $\FI$ having length $\gd$ which forces each $\gg \in {\mathcal C}$,
$\gg \le \gd$ to be a strong cardinal whose strongness is indestructible under
$\gg^+$-weakly closed partial orderings satisfying the Prikry property.
%Because each component partial ordering of $\FI^1$
Because each $\FI(\gg_\ga, \gd_\ga)$ for $\ga < \gk$ is an Easton support
iteration of length $\gd_\ga$, by the definition of $\FI$, $\FI^1$ is an
Easton support iteration of length $\gd$ which is the direct limit of its components.
Therefore, by the proof of Lemma \ref{l1},
$V \models ``\gd$ is $\gd^+$ supercompact''.
Since $V \models {\rm GCH}$ and $\gd \in {\mathcal C}$,
$V \models ``\gd$ is $2^\gd$ supercompact and strong''.
Hence, by \cite[Lemma 2.1]{AC2}, $V \models ``\gd$ is a limit of strong cardinals'',
which contradicts that $\gd \in {\mathcal C}$ and in $V$,
${\mathcal C} = \{\gd < \gk \mid \gd$ is a strong
cardinal which is not a limit of strong cardinals$\}$.
Consequently, $V^{\FI \ast \dot \FP} \models ``\gd$ is not $\gd^+$ strongly compact'',
$V^{\FI \ast \dot \FP} \models ``$Level by level
equivalence holds at $\gd$'', and $V^{\FI \ast \dot \FP} \models ``\gd$ is not
supercompact up to a measurable cardinal''.
This completes the proof of Lemma \ref{l3}.
\end{proof}
Lemmas \ref{l1} -- \ref{l3} and the intervening remarks
complete the proof of Theorem \ref{t2}.
\end{proof}
We conclude by asking whether it is possible to prove analogues to
either Theorems \ref{t1} or \ref{t2} for universes in which there are
no restrictions on the large cardinal structure.
As our methods indicate, this would require rather different proofs.
\begin{thebibliography}{99}
%\bibitem{A06a} A.~Apter, ``Failures of SCH and
%Level by Level Equivalence'', {\it Archive for
%Mathematical Logic 45}, 2006, 831--838.
\bibitem{A97} A.~Apter, ``Patterns of
Compact Cardinals'', {\it Annals of
Pure and Applied Logic 89}, 1997, 101--115.
\bibitem{AC1} A.~Apter, J.~Cummings, ``Identity Crises
and Strong Compactness'',
{\it Journal of Symbolic Logic 65}, 2000, 1895--1910.
\bibitem{AC2} A.~Apter, J.~Cummings, ``Identity Crises
and Strong Compactness II: Strong Cardinals'',
{\it Archive for Mathematical Logic 40}, 2001, 25--38.
\bibitem{AG14} A.~Apter, M.~Gitik, ``On Tall Cardinals
and Some Related Generalizations'', {\it Israel Journal of
Mathematics 202}, 2014, 343--373.
\bibitem{AS97a} A.~Apter, S.~Shelah, ``On the Strong Equality between
Supercompactness and Strong Compactness'',
{\it Transactions of the American Mathematical Society 349}, 1997, 103--128.
\bibitem{G86} M.~Gitik, ``Changing Cofinalities
and the Nonstationary Ideal'',
{\it Israel Journal of Mathematics 56},
1986, 280--314.
\bibitem{GS} M.~Gitik, S.~Shelah,
``On Certain Indestructibility of
Strong Cardinals and a Question of
Hajnal'', {\it Archive for Mathematical
Logic 28}, 1989, 35--42.
%\bibitem{H1} J.~D.~Hamkins,
%``Destruction or Preservation As You
%Like It'',
%{\it Annals of Pure and Applied Logic 91},
%1998, 191--229.
%\bibitem{H03} J.~D.~Hamkins, ``Extensions with the
%Approximation and Cover Properties have No New
%Large Cardinals'', {\it Fundamenta Mathematicae 180},
%2003, 257--277.
%\bibitem{H2} J.~D.~Hamkins, ``Gap Forcing'',
%{\it Israel Journal of Mathematics 125}, 2001, 237--252.
%\bibitem{H3} J.~D.~Hamkins, ``Gap Forcing:
%Generalizing the L\'evy-Solovay Theorem'',
%{\it Bulletin of Symbolic Logic 5}, 1999, 264--272.
\bibitem{H09} J.~D.~Hamkins, ``Tall Cardinals'',
{\it Mathematical Logic Quarterly 55}, 2009, 68--86.
%\bibitem{H4} J.~D.~Hamkins, ``The Lottery Preparation'',
%{\it Annals of Pure and Applied Logic 101},
%2000, 103--146.
\bibitem{HW} J.~D.~Hamkins, W.~H.~Woodin,
``Small Forcing Creates neither Strong nor
Woodin Cardinals'', {\it Proceedings of the
American Mathematical Society 128}, 2000, 3025--3029.
\bibitem{J} T.~Jech, {\it Set Theory.
The Third Millennium Edition,
Revised and Expanded}, Springer-Verlag,
Berlin and New York, 2003.
%\bibitem{K} A.~Kanamori, {\it The
%Higher Infinite}, second edition,
%Springer-Verlag, Berlin and New York, 2003.
\bibitem{Ke} J.~Ketonen, ``Strong Compactness and
Other Cardinal Sins'', {\it Annals of Mathematical
Logic 5}, 1972, 47--76.
%\bibitem{KM} Y.~Kimchi, M.~Magidor, ``The Independence
%between the Concepts of Compactness and Supercompactness'',
%circulated manuscript.
%\bibitem{La} P.~Larson, {\it The Stationary Tower.
%Notes on a Course by W$.$ Hugh Woodin}, {\bf University
%Lecture Series 32}, American Mathematical Society,
%Providence, 2004.
%\bibitem{L} R.~Laver, ``Making the
%Supercompactness of $\gk$ Indestructible
%under $\gk$-Directed Closed Forcing'',
%{\it Israel Journal of Mathematics 29},
%1978, 385--388.
\bibitem{LS} A.~L\'evy, R.~Solovay,
``Measurable Cardinals and the Continuum Hypothesis'',
{\it Israel Journal of Mathematics 5}, 1967, 234--248.
\bibitem{Ma76} M.~Magidor, ``How Large is the First
Strongly Compact Cardinal? or A Study on
Identity Crises'', {\it Annals of
Mathematical Logic 10}, 1976, 33--57.
%\bibitem{Me} T.~Menas, ``On Strong Compactness and
%Supercompactness'', {\it Annals of Mathematical Logic 7},
%1974, 327--359.
%\bibitem{SRK} R.~Solovay, W.~Reinhardt, A.~Kanamori,
%``Strong Axioms of Infinity and Elementary Embeddings'',
%{\it Annals of Mathematical Logic 13}, 1978, 73--116.
\end{thebibliography}
\end{document}
\begin{lemma}\label{l2}
Suppose $\gd$ is a limit of members of ${\mathcal C}$. Then
$V \models ``\gd$ is measurable'' iff $V^\FI \models ``\gd$ is measurable''.
\end{lemma}
\begin{proof}
We use the notation of Lemma \ref{l1}.
Suppose $\gd$ is a limit of members of ${\mathcal C}$.
Assume first that $V \models ``\gd$ is measurable''.
The factorization $\FI = \FI_\gd \ast \dot \FI^\gd$ and the fact
$\forces_{\FI_\gd} ``$ Forcing with $\dot \FI^\gd$ adds no bounded subsets of
$(\gd^*)^V$'' indicates it suffices to show that $\gd$ remains
measurable in $V^{\FI_\gd}$.
However, since $\gd$ is measurable iff $\gd$ is $\gd$ strongly compact iff
$\gd$ is $\gd$ supercompact, the argument given in the third paragraph of the
proof of Lemma \ref{l1} tells us that $\gd$ remains measurable in $V^{\FI_\gd}$.
Next, suppose that $V^\FI \models ``\gd$ is measurable''.
The factorization just given along with the closure property of $\FI^\gd$ mentioned
tell us that $V^{\FI_\gd} \models ``\gd$ is measurable''.
Because $\FI_\gd$ is $\gd$-c.c$.$ and $V^{\FI_\gd} \models ``\gd$ is $\gd$
strongly compact'', the argument given in the second paragraph of the proof of Lemma \ref{l1}
now yields that $V \models ``\gd$ is measurable''.
This completes the proof of Lemma \ref{l2}.
\end{proof}